Auction Design for Auction Design for Atypical SituationsAtypical Situations

OverviewOverview

General review of common auctionsGeneral review of common auctions Auction design for agents with hard Auction design for agents with hard

valuation problemsvaluation problems Auction design for goods in Auction design for goods in

unlimited supplyunlimited supply

Auction DesignAuction Design

Many different protocolsMany different protocols Major auction types:Major auction types:

Ascending Price (English)Ascending Price (English) Descending Price (Dutch)Descending Price (Dutch) First Price, Sealed BidFirst Price, Sealed Bid Second Price, Sealed Bid (Vickrey)Second Price, Sealed Bid (Vickrey)

Essentially mixing and matching Essentially mixing and matching certain properties, but some certain properties, but some combinations work better than otherscombinations work better than others

Auction EvaluationAuction Evaluation

Revenue for the sellersRevenue for the sellers Profit for the biddersProfit for the bidders Avoidance of “winner’s curse” helps Avoidance of “winner’s curse” helps

bothboth Winner in most auction protocols is the Winner in most auction protocols is the

participant who made the biggest participant who made the biggest overvaluation mistakeovervaluation mistake

Auctions that decouple an agent’s bid from Auctions that decouple an agent’s bid from the actual price paid encourage higher the actual price paid encourage higher bidsbids

Ascending Price Ascending Price (English)(English)

Most commonly known protocolMost commonly known protocol Auctioneer starts with an opening Auctioneer starts with an opening

bid and successively raises the price bid and successively raises the price as participants are willingas participants are willing

Allows for dynamic adjustment of Allows for dynamic adjustment of bidders’ valuations by giving bidders’ valuations by giving information about other biddersinformation about other bidders

Buyers bid at most their utility, Buyers bid at most their utility, which may be adjusted on the flywhich may be adjusted on the fly

Descending Price Descending Price (Dutch)(Dutch)

Biddings starts at an extremely high Biddings starts at an extremely high price and descends until someone price and descends until someone claims the itemclaims the item

High degree of Winner’s CurseHigh degree of Winner’s Curse Intuitively, raises seller’s revenue in Intuitively, raises seller’s revenue in

cases when the high bidder wants an cases when the high bidder wants an item badlyitem badly

First Price, Sealed BidFirst Price, Sealed Bid

Buyers submit bids onceBuyers submit bids once No knowledge of one another’s bidsNo knowledge of one another’s bids Auctioneer opens bids and sells the Auctioneer opens bids and sells the

item to the highest bidder at the price item to the highest bidder at the price he submittedhe submitted

Encourages buyers to bid Encourages buyers to bid conservatively (shade down from conservatively (shade down from utility) to maximize profit versus utility) to maximize profit versus probability of winningprobability of winning

Second Price, Sealed Bid Second Price, Sealed Bid (Vickrey)(Vickrey)

Designed to alleviate Winner’s Curse Designed to alleviate Winner’s Curse in the First Price, Sealed Bid protocolin the First Price, Sealed Bid protocol

Same sealed bid format, but item is Same sealed bid format, but item is awarded to the highest bidder at the awarded to the highest bidder at the second highest pricesecond highest price

Buyers can bid their utility to increase Buyers can bid their utility to increase probability of winning, but are probability of winning, but are guaranteed a price closer to market guaranteed a price closer to market consensusconsensus

Optimal Auction Design for Optimal Auction Design for Agents with Hard Valuation Agents with Hard Valuation

ProblemsProblemsDavid C ParkesDavid C Parkes

MotivationMotivation Standard auction theory assumes that Standard auction theory assumes that

either agents know own their utility for either agents know own their utility for an item (private value) or that there is a an item (private value) or that there is a common utility that is unknown to the common utility that is unknown to the agents (common value)agents (common value)

As transactions are increasingly turned As transactions are increasingly turned over to software agents, the cost of over to software agents, the cost of obtaining this utility value may be obtaining this utility value may be significantsignificant

Certain auction designs can simplify this Certain auction designs can simplify this problemproblem

Paper GoalsPaper Goals

Compare the performance of agents Compare the performance of agents with hard valuation problems within with hard valuation problems within three auction designsthree auction designs Posted Price, sequentialPosted Price, sequential Second Price, sealed bidSecond Price, sealed bid Ascending PriceAscending Price

What the paper isn’t about: Actual What the paper isn’t about: Actual methods for refining beliefs about methods for refining beliefs about valuesvalues

Examples of Hard Examples of Hard ValuationValuation

Agents bidding for components on Agents bidding for components on behalf of a manufacturerbehalf of a manufacturer

Agents bidding for collectibles or Agents bidding for collectibles or other raritiesother rarities

Problem FormulationProblem Formulation

Agents with hard valuation problems Agents with hard valuation problems operate in three phases:operate in three phases: Metadeliberation: Decide how much Metadeliberation: Decide how much

effort to spend refining the valuation of effort to spend refining the valuation of the itemthe item

Valuation: The refinement process – Valuation: The refinement process – solve an optimization problem, interact solve an optimization problem, interact with a human expert, etcwith a human expert, etc

BiddingBidding

Agent ModelAgent Model Each agent has an unknown true value Each agent has an unknown true value

for an itemfor an item Each agent maintains an upper and Each agent maintains an upper and

lower bound, in which the true value is lower bound, in which the true value is assumed to be somewhere, uniformly assumed to be somewhere, uniformly distributed, in betweendistributed, in between

Expected true value is then the Expected true value is then the average of the upper and lower boundsaverage of the upper and lower bounds

The deliberation process refines the The deliberation process refines the upper and lower boundsupper and lower bounds

Agent ModelAgent Model

Deliberation

Incurs cost C

Upper Bound

Lower Bound

New Upper Bound

New Lower Bound

Expected True Value

New ExpectedTrue Value

ParametersParameters

Cost of Deliberation (C)Cost of Deliberation (C) Computational Effectiveness of Computational Effectiveness of

Deliberation (1-Deliberation (1-))

MetadeliberationMetadeliberation

Solve the tradeoff between reducing Solve the tradeoff between reducing uncertainty and incurring the cost of uncertainty and incurring the cost of deliberationdeliberation

Deliberation is only worthwhile when:Deliberation is only worthwhile when: It changes the agent’s bidIt changes the agent’s bid The new bid has a greater expected The new bid has a greater expected

utility than the old bidutility than the old bid Metadeliberation strategies vary by Metadeliberation strategies vary by

auction typeauction type

Metadeliberation StrategiesMetadeliberation StrategiesVickrey AuctionVickrey Auction

Need distributional information about the bids Need distributional information about the bids of other agentsof other agents

Paper assumes such information is somehow Paper assumes such information is somehow obtained by the agents (eg learning)obtained by the agents (eg learning)

Metadeliberation strategy:Metadeliberation strategy: Deliberate until utility of placing a bid now is Deliberate until utility of placing a bid now is

greater than the estimated utility of placing a bid greater than the estimated utility of placing a bid after another deliberation stepafter another deliberation step

Bid utility is a nonlinear function of expected valueBid utility is a nonlinear function of expected value Higher bid decreases profit but raises probability of Higher bid decreases profit but raises probability of

winningwinning

Metadeliberation StrategiesMetadeliberation StrategiesVickrey AuctionVickrey Auction

Length of time that an agent spends Length of time that an agent spends deliberating depends on:deliberating depends on: The number of agents in the auctionThe number of agents in the auction The agent’s current upper and lower The agent’s current upper and lower

bounds on the value of the item bounds on the value of the item The computational effectiveness of The computational effectiveness of

deliberation (1-deliberation (1-)) The cost of deliberation (C)The cost of deliberation (C)

Metadeliberation StrategiesMetadeliberation StrategiesPosted Price SequentialPosted Price Sequential

No uncertainty about the actions of No uncertainty about the actions of other agentsother agents

Need only to worry about the cost of Need only to worry about the cost of the goodthe good

Deliberate only when the ask price is Deliberate only when the ask price is within the bounds of some threshold within the bounds of some threshold function function ((, C, , C, ))*

Accept Price

Deliberate

Reject Pricev

v

Metadeliberation StrategyMetadeliberation StrategyAscending PriceAscending Price

Third action available in addition to Third action available in addition to bid and deliberate: waitbid and deliberate: wait Agents that wait benefit from the Agents that wait benefit from the

deliberation of othersdeliberation of others Optimal Strategy:Optimal Strategy:

Bid

Wait, or Deliberate if auction will close, with probability 1/(Na -1)

Leave auctionv

v

Evaluation MetricsEvaluation Metrics Efficiency:Efficiency:

True Value for the Good for the Winning True Value for the Good for the Winning Agent / Maximum True Value over All Agent / Maximum True Value over All AgentsAgents

Revenue:Revenue:Price Paid for the Good / Maximum True Price Paid for the Good / Maximum True Value over All AgentsValue over All Agents

Utility of Participation:Utility of Participation:(Surplus to Winning Agent – Total (Surplus to Winning Agent – Total Deliberation Cost for All Agents) / Deliberation Cost for All Agents) / Number of AgentsNumber of Agents

Evaluation ArmsEvaluation Arms

Variance of:Variance of: Number of AgentsNumber of Agents Computational Effectiveness (1-Computational Effectiveness (1-)) Cost of Deliberation (C)Cost of Deliberation (C) Agent “experience” – Adjust C and Agent “experience” – Adjust C and for for

fractions of the agent populationfractions of the agent population

Varying the Number of Varying the Number of AgentsAgents

+ Ascending Price

X Sealed Bid

O Posted Price Sequential

Varying the Bidding Varying the Bidding IncrementIncrement

+ Ascending Price

Varying the Computational Varying the Computational Effectiveness of Effectiveness of

DeliberationDeliberation

+ Ascending Price

X Sealed Bid

O Posted Price Sequential

Varying Agent Varying Agent ExperienceExperience

+ Ascending Price

X Sealed Bid

O Posted Price Sequential

Competitive Auctions and Competitive Auctions and Digital GoodsDigital Goods

Andrew GoldbergAndrew Goldberg

Jason D HartlineJason D Hartline

Andrew WrightAndrew Wright

MotivationMotivation Looking for an optimal way to sell goods Looking for an optimal way to sell goods

in unlimited supplyin unlimited supply Downloadable musicDownloadable music Pay per view moviesPay per view movies

Examines auctions as an alternative to Examines auctions as an alternative to fixed pricing, which requires expensive fixed pricing, which requires expensive and probably inaccurate market researchand probably inaccurate market research

Scary implication: Charge more for Scary implication: Charge more for media created by entities with small, media created by entities with small, rabid followings?rabid followings?

The Optimal Threshold The Optimal Threshold FunctionFunction

Given a set of bids, determine the Given a set of bids, determine the single price that maximizes revenuesingle price that maximizes revenue

3 3 3 2 2 1

• 3 at 3 (9)• 5 at 2 (10)• 6 at 1 (6)

Sell…

Paper GoalsPaper Goals

Examine classes of single round, Examine classes of single round, sealed bid auctions for products with sealed bid auctions for products with no marginal cost of reproductionno marginal cost of reproduction

Need to solve tradeoff between selling Need to solve tradeoff between selling a lot at a low price versus a few at a a lot at a low price versus a few at a high pricehigh price

Need to ensure that participants bid Need to ensure that participants bid their utilitiestheir utilities

Would like an auction mechanism that Would like an auction mechanism that compares well to optimal fixed pricingcompares well to optimal fixed pricing

TerminologyTerminology

Truthful auctionsTruthful auctions Encourage participants to bid their utilityEncourage participants to bid their utility More formally: More formally:

Bidder’s profit (bid – price if wins, or 0 Bidder’s profit (bid – price if wins, or 0 otherwise) is maximized when bid is the same otherwise) is maximized when bid is the same as utility for any fixed values for the bids of as utility for any fixed values for the bids of other participantsother participants

Example: VickreyExample: Vickrey Counterexample: First price sealed bidCounterexample: First price sealed bid Why is this important?Why is this important?

Revenue is maximized in truthful auctionsRevenue is maximized in truthful auctions

Truthful Auction Truthful Auction ExampleExample

Imagine participating in an auction Imagine participating in an auction for a jar of 100 pennies that you (and for a jar of 100 pennies that you (and only you) have countedonly you) have counted

In a first price sealed bid auction, In a first price sealed bid auction, you cannot hope to profit by bidding you cannot hope to profit by bidding 100100

In a Vickrey auction, by bidding 100 In a Vickrey auction, by bidding 100 you will at worst break even, but you will at worst break even, but most likely pay lessmost likely pay less

TerminologyTerminology

Competitive auctionsCompetitive auctions Produce revenue within a constant Produce revenue within a constant

factor of optimal fixed pricingfactor of optimal fixed pricing Must vary the number of items sold based Must vary the number of items sold based

on the bids receivedon the bids received Why is this important?Why is this important?

Matching optimal fixed pricing is the best Matching optimal fixed pricing is the best possible resultpossible result

Being within a constant factor is a Being within a constant factor is a reasonable approximationreasonable approximation

Evaluated Auction Evaluated Auction DesignsDesigns

All auctions in this paper are single All auctions in this paper are single round, sealed bidround, sealed bid

Auction mechanisms studied:Auction mechanisms studied: Deterministic Deterministic

Deterministic Optimal ThresholdDeterministic Optimal Threshold RandomizedRandomized

Single PriceSingle Price Dual PriceDual Price Weighted PairingWeighted Pairing

Bid IndependenceBid Independence

Agent’s bid determines whether or Agent’s bid determines whether or not he wins the auction, but not the not he wins the auction, but not the priceprice

Typically multi-price, but not alwaysTypically multi-price, but not always Example: VickreyExample: Vickrey Why is this important?Why is this important?

Bid independence allows for one to bid Bid independence allows for one to bid her utility and still hope for a profither utility and still hope for a profit

Deterministic auctions must be bid Deterministic auctions must be bid independent in order to be truthfulindependent in order to be truthful

Truthful Deterministic Truthful Deterministic AuctionsAuctions

Consider the Consider the deterministic optimal deterministic optimal threshold auctionthreshold auction To determine if bid bTo determine if bid bii wins wins

Remove bRemove bii from the set of bids (ensure bid from the set of bids (ensure bid independence/truthfulness)independence/truthfulness)

Determine the threshold price at which maximal Determine the threshold price at which maximal revenue is attained in the remaining set of bidsrevenue is attained in the remaining set of bids

If bIf bii >= this threshold, accept b >= this threshold, accept bii at the threshold at the threshold priceprice

Note that this removal of bi is the only thing Note that this removal of bi is the only thing that differentiates this auction from optimal that differentiates this auction from optimal fixed pricingfixed pricing

Deterministic Optimal Deterministic Optimal Threshold Auction ExampleThreshold Auction Example

3 3 3 2 2 15

Step 1: Remove the bid to be evaluated:3

Step 2: Compute the optimal threshold on the remaining bids:

3 3 2 2 15

Sell1 at 5: 53 at 3: 95 at 2: 106 at 1: 6Optimal threshold is 2

Step 3: Compare the removed bid to the optimal threshold3 > 2

Step 4: Accept bid 3 at price 2

Truthful Deterministic Truthful Deterministic AuctionsAuctions

Theoretical ResultsTheoretical Results Truthful Deterministic Bid-Truthful Deterministic Bid-

Independent Auctions are not Independent Auctions are not Competitive in the worst caseCompetitive in the worst case

Removing bRemoving bii causes bigger problems causes bigger problems than one would expectthan one would expect

Consider an input set where the high Consider an input set where the high bid bid hh occurs occurs rr times, and there are times, and there are ((h – 1) (r – 1) h – 1) (r – 1) other bids at 1other bids at 1

Proof SketchProof Sketch

Example: Example: hh = 5, = 5, rr = 3 = 3

5 5 5 1 1 1 1 1 1 11

Sell… • 11 at price 1 (11 revenue)• 3 at price 5 (15 revenue)

Proof Sketch, ContinuedProof Sketch, Continued

Deterministic Optimal Threshold Deterministic Optimal Threshold Auction on this input:Auction on this input:

5 5 5 1 1 1 1 1 1 11

5 5 5 1 1 1 1 1 1 11

To determine if b1 wins, remove it and compute To determine if b1 wins, remove it and compute the threshold on the rest of the inputthe threshold on the rest of the input

Sell 2 units at 5, or 10 units at 1 Threshold falls to 1, so end up selling Threshold falls to 1, so end up selling

to the high bids at the low priceto the high bids at the low price

More ResultsMore Results This result generalizes to all This result generalizes to all

Deterministic Auctions due to the Deterministic Auctions due to the theorem that all Truthful Deterministic theorem that all Truthful Deterministic Auctions are Bid IndependentAuctions are Bid Independent

Proof intuition:Proof intuition: Can’t profit by bidding utility if you’ll pay it Can’t profit by bidding utility if you’ll pay it

upon winningupon winning Paper goes on to present empirical Paper goes on to present empirical

evidence that on realistic data, the worst evidence that on realistic data, the worst case for these auctions doesn’t come up case for these auctions doesn’t come up oftenoften

Random Sampling AuctionsRandom Sampling AuctionsMotivationMotivation

Auction mechanisms need not only Auction mechanisms need not only be resistant to bad inputs, but also be resistant to bad inputs, but also to attackto attack

By using some nondeterminism in By using some nondeterminism in deciding who wins and at what deciding who wins and at what price, we can avoid dominance by price, we can avoid dominance by worst case inputsworst case inputs

Random Sampling Random Sampling AuctionsAuctions

Choose a random sample from the set of bidsChoose a random sample from the set of bids Run the optimal threshold function on the Run the optimal threshold function on the

sample and use the result on the bids not in the sample and use the result on the bids not in the samplesample

Single priceSingle price NondeterministicNondeterministic Dual price variant:Dual price variant:

Choose roughly half of the bids for the sampleChoose roughly half of the bids for the sample Calculate thresholds on both setsCalculate thresholds on both sets Use the threshold from one set on the other, and vice Use the threshold from one set on the other, and vice

versaversa Avoids having to throw out bids from the sampleAvoids having to throw out bids from the sample

Random Sampling Random Sampling ExampleExample3 3 3 2 2 1

Step 1: Choose subset at random: 3 2

5

2

Step 2: Compute optimal threshold: 3 2 2

Sell 1 at 3: 3Sell 3 at 2: 6Optimal threshold is 2

Step 3: Apply optimal threshold to those not in the sample:

3 3 15

Step 4:

Accept 3 35 at price 2

Random Sampling AuctionsRandom Sampling AuctionsTheoretical ResultsTheoretical Results

Random sampling auctions are competitive Random sampling auctions are competitive Revenue generated is within a constant factor of Revenue generated is within a constant factor of

optimal fixed pricing with arbitrarily high (but not optimal fixed pricing with arbitrarily high (but not 1.0) probability1.0) probability

The higher the probability, the lower the constantThe higher the probability, the lower the constant General flavor of the proof:General flavor of the proof:

The chosen subset is a good representation of the The chosen subset is a good representation of the whole most of the timewhole most of the time

The revenue lost from losing the sampled bids is The revenue lost from losing the sampled bids is a constant factor of the wholea constant factor of the whole

The dual price version performs even betterThe dual price version performs even better

Weighted Pairing Weighted Pairing AuctionsAuctions

To determine if a particular bidder i wins To determine if a particular bidder i wins with bid bi:with bid bi: Choose another bid b with probability Choose another bid b with probability

proportional to its value –higher bids get picked proportional to its value –higher bids get picked more oftenmore often

Compare b with bi. If bCompare b with bi. If bii >= b, bidder i wins and >= b, bidder i wins and pays bpays b

Multi-priceMulti-price NondeterministicNondeterministic High bidders likely to win and pay high High bidders likely to win and pay high

prices, but some low bidders sneak in as wellprices, but some low bidders sneak in as well

Weighted Pairing Weighted Pairing ExampleExample3 3 3 2 2 15

Step 1: Choose a bid 3

Step 2: Choose another bid with probability proportional to its value

2Step 3: Compare 3 >

Step 4: Bid 3wins and pays price 2

2

Weighted Pairing AuctionsWeighted Pairing AuctionsTheoretical ResultsTheoretical Results

Weighted Pairing auctions are not Weighted Pairing auctions are not quite competitivequite competitive

Within a logarithmic factor of fixed Within a logarithmic factor of fixed pricing, so not bad eitherpricing, so not bad either

Perform well on inputs that random Perform well on inputs that random sampling does notsampling does not